Stationary layered solutions in ${\mathbb{R}}^2$ for a class of non autonomous Allen-Cahn equations

Abstract.

We consider a class of non autonomous Allen-Cahn equations

\begin{equation} -\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in {\mathbb{R}}^{2}, \end{equation}

where $W\in{\cal{C}}^{2}({\mathbb{R}},{\mathbb{R}})$ is a multiple-well potential and $a\in{\cal{C}}({\mathbb{R}},{\mathbb{R}})$ is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as $x\to\pm\infty$ corresponding to minima of W. We show, via variational methods, that under a nondegeneracy condition on the set of heteroclinic solutions of the associated ordinary differential equation $-\ddot q(x)+a(x)W'(q(x))=0,$ $x\in{\mathbb{R}},$ the equation (0.1) has solutions which depends on both the variables x andy. In contrast, when a is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on x.